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We prove that, for ρ>2, the ρ‐variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in  L p   for 1< p <∞. The analogous result holds for the  n ‐dimensional Riesz transform on  n ‐dimensional Lipschitz graphs, as well as for other singular integral operators with odd kernel. In particular, our results strengthen the classical theorem on the  L  2  boundedness of the Cauchy transform on Lipschitz graphs by Coifman, McIntosh, and Meyer.

Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs